**7.2. Angular Spreading and External shocks**

Comparison of Eqs. 29 and 30 (using
*R*_{E}
*R*_{E}) reveals that
*T*_{ang}
*T*_{radial}. As long as the shell's angular width
is larger than
^{-1},
any temporal structure that could have
arisen due to irregularities in the properties of the shell or in the
material that it encounters will be spread on a time given by
*T*_{ang}. This means that *T*_{ang} is the
minimal time scale for the observed temporal variability:
*T*
*T*_{ang}.

Comparison with the intrinsic time scales yields two cases:

(33) |

In Type-I models, the duration of the burst is determined by the emission radius and the Lorentz factor. It is independent of . This type of models include the standard "external shock model" [27, 18, 233] in which the relativistic shell is decelerated by the ISM, the relativistic magnetic wind model [226] in which a magnetic Poynting flux runs into the ISM, or the scattering of star light by a relativistic shell [234, 235].

In Type-II models, the duration of the burst is determined by the thickness of the relativistic shell, (that is by the duration that the source is active and produces the relativistic wind). The angular spreading time (which depends on the the radius of emission) is shorter and therefore irrelevant. These models include the "internal shock model" [28, 29, 30], in which different parts of the shell are moving with different Lorentz factor and therefore collide with one another. A magnetic dominated version is given by Thompson [224].

The majority of GRBs have a complex temporal structure (e.g.
section 2.2) with
*T* /
*T* of order
100. Consider a Type-I model. Angular spreading means that at any
given moment the observer sees a whole region of angular width
_{E}^{-1}. Any variability in the emission
due to different conditions in different radii on a time scale smaller than
*T*_{ang} is erased unless the
angular size of the emitting region is smaller than
_{E}^{-1}. Thus, such a source can
produce only a smooth single humped burst with
= 1 and no
temporal structure on a time-scale
*T* <
*T*. Put in other words a shell, of a Type-I model, and with an
angular width larger than
_{E}^{-1} cannot produce a variable burst
with >> 1. This is
the angular spreading problem.

On the other hand a Type-II model contains a thick shell
>
*R*_{E} /
_{E}^{2} and it can produce a variable
burst. The variability time scale, is again limited
*T* >
*T*_{ang} but now it can
be shorter than the total duration *T*. The duration of the burst
reflects the time that the "inner engine" operates. The variability
reflects the radial inhomogeneity of the shell which was
produced by the source (or the cooling time if it is longer than
/ *c*). The
observed temporal variability provides an upper limit
to the scale of the radial inhomogeneities in the shell and to the scale
in which the "inner engine" varies. This is a remarkable conclusion
in view of the fact that the fireball hides the "inner engine".

Can an external shock give rise to a Type-II behavior? This would have
been possible if we could set the parameters of the external shock
model to satisfy
*R*_{E}
2_{E}
*c* *T*. As
discussed in 8.7.1 the deceleration
radius for a thin shell with
an initial Lorentz factor
is given by

(34) |

and the observed duration is
*l* ^{-8/3} / *c*. The deceleration is
gradual and the Lorentz factor of the emitting region
_{E} is
similar to the original Lorentz factor of the shell
. It seems
that with an arbitrary large Lorentz factor
we can get a
small enough deceleration radius *R*_{E}. However, Eq. 34
is valid only for thin shells satisfying
>
*l*^{-8/3}
[233].
As
increases above a critical value
_{c}
= (*l* /
)^{3/8} the
shell can no longer be considered
thin. In this situation the reverse shock penetrating the shell becomes
ultra-relativistic and the shocked matter moves with Lorentz factor
_{E}
= _{c}
< which
is independent of the initial Lorentz factor of the shell
. The
deceleration radius is now given by *R*_{E} =
^{1/4}
*l*^{3/4}, and it is
independent of the initial Lorentz factor of the shell. The behavior
of the deceleration radius *R*_{E} and observed duration as
function of the shell Lorentz factor
is given
in Fig. 15 for a shell of thickness
= 3 ×
10^{12}cm. This emission radius *R*_{E} is always
larger than /
_{E}^{2} - thus an external shock cannot be
of type II.